Finite Group Databasebeta

← C2×#80741 #11664926

Subgroups — C2×#80741 #11664926

Order 768.

2006 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5622C212yesmaximal
2C2×C3×#467C211yesmaximal, characteristic
3C2×#5183C211yesmaximal, characteristic, complemented
4C2×Dic3×Q16C211yesmaximal, characteristic, complemented
5C2×#5116C211yesmaximal, characteristic
6C22×C8×Dic3C211yesmaximal, characteristic
7C2×#5051C212yesmaximal
8#80741C2116yesmaximal, direct product factor
9C2×#4874C211yesmaximal, characteristic, complemented
10C2×#4873C211yesmaximal, characteristic, complemented
11C2×#4792C212yesmaximal, complemented
12C2×#4790C212yesmaximal, complemented
13C2×#3170—31nomaximal, Sylow
14C22×C3×Q16C2211yescharacteristic
15C23×C3×C8C2211yescharacteristic
16C2×C3×C22⋊Q8C2212yes—
17C2×Q8×Dic3C2212yes—
18C2×#944C2212yes—
19C22×#216C2212yes—
20C2×#924C2212yes—
21C22×C4×Dic3C2211yescharacteristic
22C23×C3⋊C8C2211yescharacteristic
23C22×Dic12C2211yescharacteristic
24#5622—28no—
25#5622C22116yes—
26C2×#889—22no—
27C2×#888C2212yes—
28C2×#887—22no—
29C2×#887C2212yes—
30C2×#878—22no—
31C2×#877—22no—
32C3×#467C2218yes—
33C2×C3×#107C2211yescharacteristic
34C2×C3×#103C2212yes—
35C2×#835C2212yes—
36#5183C2218yescomplemented
37Dic3×Q16C2218yescomplemented
38#5116C2218yes—
39C2×#818C2211yescharacteristic
40C2×#816C2212yes—
41C2×C8×Dic3C2211yescharacteristic
42C2×C8×Dic3C2211yescharacteristic
43C2×C8×Dic3C2218yes—
44C2×C8×Dic3C2211yescharacteristic
45C2×C8×Dic3C2211yescharacteristic
46#5051C22116yes—
47C2×#810C2212yes—
48C2×#807C2212yes—
49#4874C2218yescomplemented
50#4873C2218yescomplemented
51#4792C22116yescomplemented
52#4790C22116yescomplemented
53C2×#533—32no—
54#3170—316no—
55C2×#500—31no—
56C2×#494—31no—
57C2×#487—32no—
58C2×#467—31no—
59C2×#467—32no—
60C2×#467D311yescharacteristic
61C2×C4×Q16—31no—
62C2×C4×Q16—32no—
63C2×C4×Q16—31no—
64C22×C4×C8—31no—
65C22×C3×Q8C2312yes—
66C23×C3×C4C2311yescharacteristic
67C23×Dic3D411yescharacteristic
68C23×Dic3D411yescharacteristic
69C22×Dic6—22no—
70C22×Dic6C2312yes—
71C2×C3×Q16—24no—
72C2×C3×Q16C2314yes—
73C22×C3×C8—24no—
74C22×C3×C8C2311yescharacteristic
75C22×C3×C8C2314yes—
76C22×C3×C8C2311yescharacteristic
77C3×C22⋊Q8—24no—
78C3×C22⋊Q8C2318yes—
79C2×C3×C4⋊C4—22no—
80C2×C3×C4⋊C4C2312yes—
81C2×C3×C22⋊C4—22no—
82C2×#217—22no—
83Q8×Dic3—24no—
84Q8×Dic3C2318yes—
85#944—24no—
86#944C2318yes—
87C2×#216—28no—
88C2×#216C2318yes—
89C2×C4⋊Dic3—22no—
90C2×C4⋊Dic3C2312yes—
91#924—24no—
92#924C2318yes—
93C2×Dic3⋊C4—22no—
94C2×Dic3⋊C4—22no—
95C2×Dic3⋊C4—22no—
96C2×Dic3⋊C4—22no—
97C2×Dic3⋊C4—22no—
98C2×C4×Dic3—22no—
99C2×C4×Dic3—22no—
100C2×C4×Dic3D411yescharacteristic
101C2×C4×Dic3D411yescharacteristic
102C2×C4×Dic3D414yes—
103C2×C4×Dic3C2311yescharacteristic
104C2×C4×Dic3C2311yescharacteristic
105C2×C4×Dic3C2314yes—
106C22×C3⋊C8—24no—
107C22×C3⋊C8C2311yescharacteristic
108C22×C3⋊C8C2314yes—
109C22×C3⋊C8C2311yescharacteristic
110C2×Dic12—24no—
111C2×Dic12C2314yes—
112#889—216no—
113#888—24no—
114#888C2318yes—
115#887—216no—
116#887—24no—
117#887C2318yes—
118#878—216no—
119#877—216no—
120C3×#107C2314yes—
121C3×#103C2318yes—
122#835C2318yes—
123#818C2314yes—
124#816C2318yes—
125C8×Dic3C2314yes—
126C8×Dic3C2314yes—
127C8×Dic3C2314yes—
128C8×Dic3C2314yes—
129#810C2318yes—
130#807C2318yes—
131C22×Q16—32no—
132C22×Q16—31no—
133C22×Q16C2×D311yescharacteristic
134C23×C8—31no—
135C23×C8C2×D311yescharacteristic
136#533—316no—
137#533—68no—
138C2×C4⋊Q8—32no—
139C2×C4⋊Q8—32no—
140C2×#125—62no—
141C2×C22⋊Q8—32no—
142C2×C22⋊Q8—62no—
143C2×C22⋊Q8C2×D312yes—
144C2×C4×Q8—32no—
145C2×C4×Q8—32no—
146C2×C4×Q8—62no—
147C2×#117—62no—
148C22×C42—31no—
149#500—38no—
150#494—38no—
151#487—316no—
152#467—38no—
153#467—316no—
154#467C2×D318yes—
155C4×Q16—38no—
156C4×Q16—38no—
157C4×Q16—316no—
158C2×#107—31no—
159C2×#107—32no—
160C2×#107C2×D311yescharacteristic
161C2×#103—32no—
162C2×#103—32no—
163C2×#103—32no—
164C2×#103C2×D312yes—
165C2×C4×C8—31no—
166C2×C4×C8—31no—
167C2×C4×C8—38no—
168C2×C4×C8—31no—
169C2×C4×C8—31no—
170C24×C3C2×D411yescharacteristic
171C2×C3×Q8—22no—
172C2×C3×Q8—24no—
173C2×C3×Q8—24no—
174C2×C3×Q8C2414yes—
175C22×C3×C4—22no—
176C22×C3×C4—22no—
177C22×C3×C4—22no—
178C22×C3×C4—22no—
179C22×C3×C4C2×D411yescharacteristic
180C22×C3×C4C2×D412yes—
181C22×C3×C4C2412yes—
182C22×C3×C4C2411yescharacteristic
183C22×Dic3—24no—
184C22×Dic3—22no—
185C22×Dic3—22no—
186C22×Dic3Q1618yescomplemented
187C22×Dic3C2×D414yes—
188C22×Dic3C2×D411yescharacteristic
189C22×Dic3C2×D411yescharacteristic
190C22×Dic3C2×D411yescharacteristic
191C22×Dic3C2×D414yes—
192C22×Dic3C2×D411yescharacteristic
193C2×Dic6—24no—
194C2×Dic6—24no—
195C2×Dic6—22no—
196C2×Dic6—28no—
197C2×Dic6—48no—
198C2×Dic6C2414yes—
199C3×Q16—28no—
200C2×C3×C8—22no—
201C2×C3×C8—28no—
202C2×C3×C8#4114yes—
203C2×C3×C8C2412yes—
204C2×C3×C8C2412yes—
205C3×C4⋊C4—22no—
206C3×C4⋊C4—28no—
207C3×C4⋊C4C2414yes—
208C3×C22⋊C4—28no—
209#217—28no—
210#216—216no—
211C4⋊Dic3—22no—
212C4⋊Dic3—28no—
213C4⋊Dic3C2414yes—
214Dic3⋊C4—28no—
215Dic3⋊C4—28no—
216Dic3⋊C4—28no—
217Dic3⋊C4—28no—
218Dic3⋊C4—28no—
219C4×Dic3—28no—
220C4×Dic3—28no—
221C4×Dic3C2×D414yes—
222C4×Dic3C2×D412yes—
223C4×Dic3C2×D412yes—
224C4×Dic3C2×D414yes—
225C4×Dic3C2412yes—
226C4×Dic3C2412yes—
227C2×C3⋊C8—22no—
228C2×C3⋊C8—28no—
229C2×C3⋊C8#4114yes—
230C2×C3⋊C8C2412yes—
231C2×C3⋊C8C2412yes—
232Dic12—28no—
233C22×Q8—32no—
234C22×Q8—62no—
235C22×Q8C22×D312yes—
236C23×C4—31no—
237C23×C4—31no—
238C23×C4C22×D311yescharacteristic
239C2×Q16—24no—
240C2×Q16—38no—
241C2×Q16—34no—
242C2×Q16—68no—
243C2×Q16—64no—
244C2×Q16C22×D314yes—
245C22×C8—24no—
246C22×C8—31no—
247C22×C8—31no—
248C22×C8—34no—
249C22×C8—64no—
250C22×C8C22×D311yescharacteristic
251C22×C8C22×D314yes—
252C22×C8C22×D311yescharacteristic
253C4⋊Q8—38no—
254C4⋊Q8—38no—
255C4⋊Q8—64no—
256C4⋊Q8—64no—
257#125—616no—
258C22⋊Q8—24no—
259C22⋊Q8—38no—
260C22⋊Q8—616no—
261C22⋊Q8—64no—
262C22⋊Q8C22×D318yes—
263C4×Q8—38no—
264C4×Q8—38no—
265C4×Q8—64no—
266C4×Q8—616no—
267C4×Q8—64no—
268#117—616no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—32no—
271C2×C4⋊C4—62no—
272C2×C4⋊C4—62no—
273C2×C4⋊C4—62no—
274C2×C4⋊C4—62no—
275C2×C4⋊C4—62no—
276C2×C4⋊C4—62no—
277C2×C4⋊C4C22×D312yes—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—62no—
280C2×C42—31no—
281C2×C42—31no—
282C2×C42—34no—
283C2×C42—31no—
284C2×C42—31no—
285C2×C42—34no—
286C2×C42—62no—
287C2×C42—62no—
288#107—34no—
289#107—38no—
290#107C22×D314yes—
291#103—38no—
292#103—38no—
293#103—38no—
294#103C22×D318yes—
295C4×C8—34no—
296C4×C8—34no—
297C4×C8—34no—
298C4×C8—34no—
299C23×C3—22no—
300C23×C3—22no—
301C23×C3C2×Q1614yes—
302C23×C3C22×D412yes—
303C23×C3C22×D411yescharacteristic
304C3×Q8—24no—
305C3×Q8—24no—
306C3×Q8—44no—
307C2×C3×C4—22no—
308C2×C3×C4—24no—
309C2×C3×C4—22no—
310C2×C3×C4—22no—
311C2×C3×C4—24no—
312C2×C3×C4—24no—
313C2×C3×C4—44no—
314C2×C3×C4—44no—
315C2×C3×C4C22×D412yes—
316C2×C3×C4C22×D411yescharacteristic
317C2×C3×C4C2×#4112yes—
318C2×C3×C4C2×#4112yes—
319C2×C3×C4C2511yescharacteristic
320C2×Dic3—22no—
321C2×Dic3—28no—
322C2×Dic3—24no—
323C2×Dic3—24no—
324C2×Dic3—44no—
325C2×Dic3—44no—
326C2×Dic3C2×Q1614yes—
327C2×Dic3C2×Q1614yes—
328C2×Dic3C2×Q16116yescomplemented
329C2×Dic3#13514yes—
330C2×Dic3C22×D412yes—
331C2×Dic3C22×D412yes—
332C2×Dic3C22×D412yes—
333C2×Dic3C22×D412yes—
334Dic6—24no—
335Dic6—24no—
336Dic6—44no—
337Dic6—416no—
338C3×C8—22no—
339C3×C8C2×#4114yes—
340C3⋊C8—22no—
341C3⋊C8C2×#4114yes—
342C24D3×D411yescharacteristic
343C2×Q8—22no—
344C2×Q8—24no—
345C2×Q8—24no—
346C2×Q8—34no—
347C2×Q8—68no—
348C2×Q8—62no—
349C2×Q8—64no—
350C2×Q8—64no—
351C2×Q8—128no—
352C2×Q8C23×D314yes—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—38no—
358C22×C4—34no—
359C22×C4—31no—
360C22×C4—31no—
361C22×C4—31no—
362C22×C4—34no—
363C22×C4—31no—
364C22×C4—64no—
365C22×C4—62no—
366C22×C4—62no—
367C22×C4D3×D412yes—
368C22×C4D3×D411yescharacteristic
369C22×C4C23×D312yes—
370C22×C4C23×D311yescharacteristic
371Q16—28no—
372Q16—616no—
373Q16—68no—
374C2×C8—22no—
375C2×C8—28no—
376C2×C8—34no—
377C2×C8—32no—
378C2×C8—32no—
379C2×C8—68no—
380C2×C8—62no—
381C2×C8#23714yes—
382C2×C8C23×D312yes—
383C2×C8C23×D312yes—
384C4⋊C4—28no—
385C4⋊C4—22no—
386C4⋊C4—34no—
387C4⋊C4—68no—
388C4⋊C4—68no—
389C4⋊C4—68no—
390C4⋊C4—68no—
391C4⋊C4—68no—
392C4⋊C4—62no—
393C4⋊C4—68no—
394C4⋊C4C23×D314yes—
395C22⋊C4—28no—
396C22⋊C4—68no—
397C42—34no—
398C42—32no—
399C42—34no—
400C42—32no—
401C42—32no—
402C42—32no—
403C42—68no—
404C42—68no—
405C22×C3—28no—
406C22×C3—22no—
407C22×C3—22no—
408C22×C3#53212yes—
409C22×C3C22×Q1614yes—
410C22×C3C22×Q1612yes—
411C22×C3C2×#13512yes—
412C22×C3C23×D411yescharacteristic
413C3×C4—22no—
414C3×C4—44no—
415C3×C4—44no—
416C3×C4#53212yes—
417C3×C4C22×#4111yescharacteristic
418C3×C4C22×#4111yescharacteristic
419Dic3—22no—
420Dic3—44no—
421Dic3—44no—
422Dic3C22×Q1614yes—
423Dic3C22×Q1614yes—
424Dic3C2×#13514yes—
425C23—22no—
426C23—22no—
427C23D3×Q1614yes—
428C23C2×D3×D412yes—
429C23C2×D3×D411yescharacteristic
430Q8—24no—
431Q8—24no—
432Q8—44no—
433Q8—64no—
434Q8—64no—
435Q8—1216no—
436Q8—124no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—32no—
444C2×C4—32no—
445C2×C4—32no—
446C2×C4—34no—
447C2×C4—34no—
448C2×C4—316no—
449C2×C4—34no—
450C2×C4—32no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—68no—
454C2×C4—62no—
455C2×C4—64no—
456C2×C4—64no—
457C2×C4—124no—
458C2×C4—124no—
459C2×C4C2×D3×D412yes—
460C2×C4C2×D3×D411yescharacteristic
461C2×C4C2×#23712yes—
462C2×C4D3×#4112yes—
463C2×C4C24×D311yescharacteristic
464C8—22no—
465C8—34no—
466C8—62no—
467C8C2×#23714yes—
468C2×C3—24no—
469C2×C3#317014yescomplemented
470C2×C3C2×#53211yescharacteristic
471C2×C3C23×Q1611yescharacteristic
472C2×C3C22×#13511yescharacteristic
473C22—28no—
474C22—22no—
475C22—22no—
476C22#559612yes—
477C22C2×D3×Q1614yes—
478C22C2×D3×Q1612yes—
479C22D3×#13512yes—
480C22C22×D3×D411yescharacteristic
481C4—22no—
482C4—34no—
483C4—34no—
484C4—34no—
485C4—44no—
486C4—44no—
487C4—62no—
488C4—124no—
489C4—124no—
490C4#559612yes—
491C4C22×#23711yescharacteristic
492C4C2×D3×#4111yescharacteristic
493C3C2×#317011yescharacteristic, complemented, Sylow
494C2—24no—
495C2#8074114yesdirect product factor
496C2C2×#559611yescharacteristic
497C2C22×D3×Q1611yescharacteristic
498C2C2×D3×#13511yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC2×C3×#467C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenterC2×#467D3
hypocenterC3C2×#3170
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#467

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. C2×#5183C2semidirect
4. C2×Dic3×Q16C2semidirect
8. #80741C2direct (N × H)
9. C2×#4874C2semidirect
10. C2×#4873C2semidirect
11. C2×#4792C2semidirect
12. C2×#4790C2semidirect
36. #5183C22semidirect
37. Dic3×Q16C22semidirect
49. #4874C22semidirect
50. #4873C22semidirect
51. #4792C22semidirect
52. #4790C22semidirect
186. C22×Dic3Q16semidirect
328. C2×Dic3C2×Q16semidirect
469. C2×C3#3170semidirect
493. C3C2×#3170semidirect
495. C2#80741direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#56222
C2×C3×#4672
C2×#51832
C2×Dic3×Q162
C2×#51162
C22×C8×Dic32
C2×#50512
#807412
C2×#48742
C2×#48732
C2×#47922
C2×#47902
C2×#31703